Question

$|a| = \sqrt{3}, |b| = 5, c = 10$ and angle between $b$ and $c$ is $\frac{\pi}{3}$. If $a$ is perpendicular to $b \times c$, then the value of $|a \times (b \times c)|$ is

• A. $10 \sqrt{3}$
• B. 15
• C. 30
• D. 10

Answer 1

75

Answer 2

We are given: $|a|=\sqrt{3}, |b|=5, |c|=10$, and the angle between $b$ and $c$ is $\frac{\pi}{3}$.

Also, since $a \perp (b \times c)$, it follows that the angle between $a$ and $(b \times c)$ is $90^\circ$. Hence,

$|a\times (b\times c)|=|a|\,|b\times c|\quad\text{(since } \sin 90^\circ = 1 \text{)}$.

First, compute:

$|b\times c|=|b||c|\sin\frac{\pi}{3}=5\cdot 10\cdot\frac{\sqrt{3}}{2}=50\cdot\frac{\sqrt{3}}{2}=25\sqrt{3}$.

Thus,

$|a\times (b\times c)|=\sqrt{3}\cdot 25\sqrt{3}=25\cdot3=75$.

Therefore, the value of $|a\times (b\times c)|$ is 75.